Catalog of electroweak scalar manifolds

TL;DR

The work addresses how local Higgs-sector data in HEFT cannot fix the global topology of the electroweak scalar manifold. By enforcing a smooth $G=SU(2)_L\times U(1)_Y$ action with a $V\cong S^3$ vacuum and minimal field content, it employs cohomogeneity-one classification, orbit-space analysis, and Mostert-type constructions to enumerate all admissible 4D target manifolds $M$ and their group actions, yielding 11 distinct manifolds (including $S^4$, $\mathbb{R}P^4$, $\mathbb{C}P^2$, and connected sums) with explicit singular isotropy data. The paper then analyzes local consequences (fixed-point and partial symmetry restoration) and global consequences (topological defects via $\pi_n(M)$), showing how topology can give rise to strings, monopoles, instantons, and textures, potentially observable in cosmology or high-energy experiments. It also discusses bottom-up EFT viability on all manifolds and outlines UV completions where $M$ arises from spontaneous symmetry breaking of a larger group. Overall, the results extend the HEFT landscape by mapping the full spectrum of scalar-manifold topologies compatible with the electroweak symmetry and by linking topology to phenomenology and possible UV completions.

Abstract

The local structure of the Higgs sector around the vacuum does not uniquely determine its global properties. Most of the current experimental data provides only local information, which allows for a rich variety of global features, including several distinct topologies of the scalar manifold, and the existence of zero, one, or two fixed points of the symmetry transformations. Here, I provide, under general conditions, a complete classification of realizations of the electroweak symmetry with minimal field content -- the three would-be Goldstone bosons and the Higgs -- and outline some of their physical consequences.

Figures (2)

• Figure 1: The seven possible actions of $U(1)$ on 2-dimensional surfaces. Samples of the circle ($S^1$) principal orbits are shown as thicker blue lines, while the fixed points are displayed as red dots. The red circles correspond to singular circle orbits in which opposite points should be identified. Some actions of the electroweak symmetry group on 4-dimensional surfaces can be obtained by replacing the principal orbits by 3-spheres and the singular ones by projective planes $\mathbb{R}P^3$, which is given by identifying opposite sides in a 3-sphere.
• Figure 2: Schematic representation of closed paths (in gray) with topological charge $Q = 1$ in manifolds with different fundamental groups. Blue and red circles represent $S^3$ and $S^3/\mathbb{Z}_2 \cong \mathbb{R}P^3$ orbits, respectively.